Find the Subgroup of $\mathbb Z_4 \times \mathbb Z_2$ that is not the form of $ H \times K$, where $H$ is a subgroup of $\mathbb Z_4$ and $ K$ is a subgroup of $\mathbb Z_2$
Order elements of $\mathbb Z_4 \times Z_2$ are
- $|(0 , 1)|= 2 ,|(2 ,0)|= 2,|(2 ,1)| = 2, |(3,0)| = 4, |(3,1)| = 4 , |(1,0)| = 4 ,|(1 , 1)| = 4 $
So by Lagrange`s Theorem possible order of a non trivial subgroups are $2 ,4$
Thus all the subgroups of order $2$ are
- $H_1 = \{ (0,0) ,((0,1) \} , H_2 = \{ (0,0) , (2 , 0) \} , H_3 = \{ (0 ,0) , (2 ,1) \} $ ,
Thus all the subgroups of order $4$ are
- $K_1 = < (3,0) > = \{(0,0) , (3,0), (2,0) , (1,0)\} , K_2 = < (3,1) > , K_3 = < (1 ,0)> , K_4 = <(1 ,1) > , K_4 = \{ (0 ,0) ,((0,1) , (2 ,0) , (2 ,1) \}$
I think there are no non trivial subgroup other than these Subgroups.
I think there is no Subgroup of this form of $ H \times K$, where $H$ is a subgroup of $\mathbb Z_4$ and $ K$ is a subgroup of $\mathbb Z_2$.
Is there exist a group $G_1 \times G_2$ such that $G_1 \times G_2$ has a subgroup of the form $ H \times K$, where $H$ is a subgroup of $G_1$ and $ K$ is a subgroup of $G_2$